Product of Ideals in Ring that is not UFD

Question

can someone tell me if I calculated the following products of Ideals correctly?

$I=(2,1+\sqrt{-5}),\ J=(3,1+\sqrt{-5}), K=(3,1-\sqrt{-5})$ in $\mathbb{Z}[\sqrt{-5}]$.

then

$I\cdot J=(1+\sqrt{-5})\\ I\cdot K=(1-\sqrt{-5})\\I^2=(2)\\J\cdot K=(3) $

Answer 1

As reference, this website is NOT for checking calculations. You should really learn how to use a calculator for that. I would like to direct you to Sage or Magma.

sage: K.<r> = NumberField(x^2 + 5)
sage: Ok = K.ring_of_integers() 
sage: r*r
-5
sage: I = Ideal(2, 1 + r)
sage: J = Ideal(3, 1 + r)
sage: K = Ideal(3, 1 - r)
sage: I*J
Fractional ideal (r + 1)
sage: I*K
Fractional ideal (-r + 1)
sage: I*I
Fractional ideal (2)
sage: J*K
Fractional ideal (3)
sage: 

So they are all correct.

Answer 2

You ought to know that for finitely generated ideals, for example $I = (a,b)$ and $J=(c,d)$, the ideal $IJ = (ac,ad,bc,bd)$ is the ideal generated by all products of generators. That said, you only need to check that you computed those products correctly.

For example, $$IJ=(2,1+\sqrt{-5})(3,1+\sqrt{-5}) = (6,2+2\sqrt{-5},3+3\sqrt{-5},-4+2\sqrt{-5}) \\= (2+2\sqrt{-5},3+3\sqrt{-5})=(1+\sqrt{-5}),$$ since $$3+3\sqrt{-5}-(2+2\sqrt{-5}) = 1+\sqrt{-5}.$$ The others are checked similarly.